I came across such strengthening of Cauchy's Integral Theorem:
Let $\Gamma$ be a rectifiable Jordan curve which is a border of an open set $G$. Let $f$ be continuous on $G\cup \Gamma$ and holomorphic on $G$. Then $$ \oint_\Gamma f(z) dz = 0 $$
It is relatively easy to prove this for smooth $f$ (one way is to construct a smooth homotopy that "tucks" $f$ into $G$ and prove that the integral changes continiously. Another approach is to prove the theorem for a star domain, by using linear contraction as homotopy, and then prove that every smooth contour can be decomposed into a finite number of star domains). However those approaches seem to fail if we only have rectifiability of $\Gamma$. I tried using the fact that $\Gamma$ has derivatives almost everywhere, but it wasn't of much help.
So the question is if the statement holds true under such conditions, and if not what is the counter example?